QCSP monsters and the demise of the Chen Conjecture

We give a surprising classification for the computational complexity of the Quantified Constraint Satisfaction Problem over a constraint language $\Gamma$, QCSP$(\Gamma)$, where $\Gamma$ is a finite language over $3$ elements which contains all constants. In particular, such problems are either in P, NP-complete, co-NP-complete or PSpace-complete. Our classification refutes the hitherto widely-believed Chen Conjecture. Additionally, we show that already on a 4-element domain there exists a constraint language $\Gamma$ such that QCSP$(\Gamma)$ is DP-complete (from Boolean Hierarchy), and on a 10-element domain there exists a constraint language giving the complexity class $\Theta_{2}^{P}$. Meanwhile, we prove the Chen Conjecture for finite conservative languages $\Gamma$. If the polymorphism clone of $\Gamma$ has the polynomially generated powers (PGP) property then QCSP$(\Gamma)$ is in NP. Otherwise, the polymorphism clone of $\Gamma$ has the exponentially generated powers (EGP) property and QCSP$(\Gamma)$ is PSpace-complete.